Problem: $\int (2 x^3 +2 x -3)\,dx=$ $+C$
Solution: We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int (2 x^3 +2 x -3)\,dx= 2\int x^3\,dx +2\int x\,dx -3\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int (2 x^3 +2 x -3)\,dx \\\\ &= 2\int x^3\,dx +2\int x\,dx -3\int 1\,dx \\\\ &=2 \dfrac{x^4}{4} +2\dfrac{x^2}{2} -3\dfrac{x^1}{1}+C \\\\ &=\dfrac{1}{2} x^4 +x^2 -3 x+C \end{aligned}$ In conclusion, $\int (2 x^3 +2 x -3)\,dx=\dfrac{1}{2} x^4 +x^2 -3 x+C$